A strong first-order EWPT could occur if there is a sufficiently high and wide potential barrier separating the two degenerate vacua of the thermal effective potential at critical temperature.
Introducing an extra bosonic degree of freedom could enhance the barrier and thus make the EWPT stronger.
In our model, there are two contributions to the barrier in the effective potential [1]}: one is the tree-level barrier coming from the cubic term of the tree-level potential; the other one arises from the bosonic thermal corrections to the potential at the one-loop level. To see the latter, one can expand the integrations (REF ) in the high temperature limit, i.e., \(z\equiv M_i^2(h,s)/T^2\ll 1\) , and find that there is a term proportional to \(z^{3/2}\)  that leads to terms cubic in both \(h\)  and \(s\) .
Most importantly, the barrier term cubic in \(h\)  is proportional to the Higgs portal coupling \(\lambda _m\) ,
and thus to the mixing angle \(\theta \)  (see Eq. ()) [2]}, [3]}, [4]}, [5]}.
Therefore a large value of mixing angle could strengthen the EWPT by boosting the potential barrier.
The potential barrier is shallow in the SM and its EWPT is confirmed to be a crossover [6]}.
